Geodesic
Good formal matrix rings over residue class rings
Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 85 (2023), pp. 32-42 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $p$ be a prime number, $m, n$ be natural and $m\geqslant n>0$. Let the formal matrix ring $\begin{pmatrix} \mathbf{Z}/p^m\mathbf{Z} & \mathbf{Z}/p^n\mathbf{Z}\\ \mathbf{Z}/p^n\mathbf{Z} & \mathbf{Z}/p^n\mathbf{Z} \end{pmatrix}$ be isomorphic to the endomorphism ring $E((\mathbf{Z}/p^m\mathbf{Z})\oplus (\mathbf{Z}/p^n\mathbf{Z}))$, may be of interest in data encryption. We will show that the ring $E((\mathbf{Z}/p^m\mathbf{Z})\oplus (\mathbf{Z}/p^n\mathbf{Z}))$, $m\geqslant n$, is $2$-good and $2$-nil-good for $p > 2$ and not good for $p = 2$ and $m > n$.
Keywords: ring, good ring, Morita context ring, endomorphism ring of abelian group. 
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Ts. D. Norbosambuev. Good formal matrix rings over residue class rings. Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 85 (2023), pp. 32-42. http://geodesic.mathdoc.fr/item/VTGU_2023_85_a2/

[1] G. M. Bergman, “Some examples in PI ring theory”, Israel Journal of Mathematics, 18 (1974), 257–277 | DOI | MR | Zbl

[2] J. J. Climent, P. R. Navarro, L. Tortosa, “On arithmetic of endomorphism ring $\mathrm{End}\,(\mathbb{Z}_p\times\mathbb{Z}_{p^2})$”, Applicable Algebra in Engineering, Communication and Computing, 22 (2011), 91–108 | DOI | MR | Zbl

[3] J. J. Climent, P. R. Navarro, L. Tortosa, “Key exchange protocols over noncommutative rings. The case of $\mathrm{End}\,(\mathbb{Z}_p\times\mathbb{Z}_{p^2})$”, International Journal of Computer Mathematics, 89 (2012), 1753–1763 | DOI | MR | Zbl

[4] J. J. Climent, P. R. Navarro, L. Tortosa, “An extension of the noncommutative Bergman's ring with a large number of noninvertible elements”, Applicable Algebra in Engineering, Communication and Computing, 25 (2014), 347–361 | DOI | MR | Zbl

[5] P. A. Krylov, A. A. Tuganbaev, “Formalnye matritsy i ikh opredeliteli”, Fundamentalnaya i prikladnaya matematika, 2014, no. 1 (19), 65–119

[6] P. Krylov, A. Tuganbaev, Formal matrices, Algebra and Applications, 23, Springer, 2017 | DOI | Zbl

[7] P. A. Krylov, “Opredeliteli obobschennykh matrits poryadka 2”, Fundamentalnaya i prikladnaya matematika, 2015, no. 5 (20), 95–112

[8] A. Yu. Stepanova, E. A. Timoshenko, “Matrichnoe predstavlenie endomorfizmov primarnykh grupp malykh rangov”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2021, no. 74, 30–42 | DOI | Zbl

[9] K. Morita, “Duality for modules and its applications to the theory of rings with minimum condition”, Science Reports of Tokyo Kyoiku Daigaku Section A, 6 (1958), 83–142 | MR | Zbl

[10] P. Loustaunau, J. Shapiro, “Morita contexts”, Non-Commutative Ring Theory, Lecture Notes in Mathematics, 1448, Springer, 1990, 80–92 | DOI | MR

[11] P. A. Krylov, Ts. D. Norbosambuev, “Avtomorfizmy algebr formalnykh matrits”, Sibirskii matematicheskii zhurnal, 59:5 (2018), 1116–1127 | DOI | MR | Zbl

[12] P. A. Krylov, Ts. D. Norbosambuev, “Gruppa avtomorfizmov odnogo klassa algebr formalnykh matrits”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2018, no. 53, 16–22 | DOI

[13] P. A. Krylov, “Ob izomorfizme kolets obobschennykh matrits”, Algebra i logika, 47:4 (2008), 456–463 | DOI | MR | Zbl

[14] Ts. D. Norbosambuev, “Rang formalnoi matritsy. Sistema formalnykh lineinykh uravnenii. Deliteli nulya”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2018, no. 52, 5–13 | DOI | MR

[15] Ts. D. Norbosambuev, “O summakh diagonalnykh i obratimykh obobschennykh matrits”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2015, no. 36, 34–41 | DOI

[16] Ts. D. Norbosambuev, E. A. Timoshenko, “O k-nil-khoroshikh koltsakh formalnykh matrits”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2022, no. 77, 17–26 | DOI | MR | Zbl

[17] G. Calugareanu, T. Y. Lam, “Fine rings: A new class of simple rings”, J. Algebra Appl., 15:9 (2016), 1650173 | DOI | MR | Zbl

[18] P. Danchev, “Nil-good unital rings”, Int. J. Algebra, 10:5 (2016), 239–252 | DOI

[19] M. Henriksen, “Two classes of rings generated by their units”, J. Algebra, 31 (1974), 182–193 | DOI | MR | Zbl